Best Known (77, 90, s)-Nets in Base 3
(77, 90, 29526)-Net over F3 — Constructive and digital
Digital (77, 90, 29526)-net over F3, using
- net defined by OOA [i] based on linear OOA(390, 29526, F3, 13, 13) (dual of [(29526, 13), 383748, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(390, 177157, F3, 13) (dual of [177157, 177067, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 177159, F3, 13) (dual of [177159, 177069, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(390, 177159, F3, 13) (dual of [177159, 177069, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(390, 177157, F3, 13) (dual of [177157, 177067, 14]-code), using
(77, 90, 59053)-Net over F3 — Digital
Digital (77, 90, 59053)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(390, 59053, F3, 3, 13) (dual of [(59053, 3), 177069, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(390, 177159, F3, 13) (dual of [177159, 177069, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 3-folding [i] based on linear OA(390, 177159, F3, 13) (dual of [177159, 177069, 14]-code), using
(77, 90, large)-Net in Base 3 — Upper bound on s
There is no (77, 90, large)-net in base 3, because
- 11 times m-reduction [i] would yield (77, 79, large)-net in base 3, but